This argument of Mochizuki doesn’t make sense to me

I made this point in a comment at Not Even Wrong, but I think it worth amplifying. It is regarding section 2.3 of a recent document released by Mochizuki, which I reproduce below the fold, along with my discussion. I have taken a screenshot as I don’t know if the document will be updated or not. I’m trying to figure this out as I go, it’s just so bizarre [Edit as in, I was literally thinking it through as I wrote this, and I didn’t edit it, on purpose. It’s what David Butler tags as #trymathslive on Twitter].

Here is the relevant section. My discussion below will be performed using standard notation and terminology, which clearly doesn’t affect my conclusions, and makes things easier to read.

The argument in (ii) seems to me to be the following: if we work in a skeleton of the category of topological spaces, where all one-element topological spaces are actually equal (because there is only one) then the interval [0,1] is collapsed to the circle S^1\simeq [0,1]/(0\sim 1). But doesn’t this argument also imply that every topological space is collapsed to a single point? Because, given a topological space X with at least two points x,y\in X, we just run the argument in (ii) with the obvious substitutions, so that x and y are “identified” (I will explain the quotation marks below). But x and y are arbitrary, so that all pairs of points are “identified”, and so X is replaced by the (unique) single element space, the quotient by the resulting equivalence relation. As a result, every non-empty topological space is isomorphic to every other one, in particular to the one-element space. This would seem to imply that the skeleton of the category of topological spaces collapses to the interval category with two objects, and one non-identity arrow.

The only way I can try to resolve this is that there is a purely linguistic sleight of hand here. Clearly working in a skeleton of the category of topological spaces in no way forces all non-empty topological spaces to collapse. But the use of the word “identified” here is ambiguous, and has two meanings in the current discussion:

  1. We can say in linear algebra that we identify any real n-dimensional vector space with \mathbb{R}^n, meaning that up to isomorphism, we can assume we are working with the latter. There may be some minor details that need care, like whether the canonical basis of \mathbb{R}^n is being implicitly used or not, compared to a generic vector space without a chosen basis. This is mathematical, and well-known.
  2. But we can also say that given a topological space, we identify a pair of points in that space, meaning that we perform a quotient construction. This changes the space we are working with completely, resulting with something not isomorphic to the original (this is the point of 2.3.ii), ignoring the issue of skeleta).

But the verb “identify” here is overloaded, and doing different things in each situation. This is obvious to any mathematician. And the complaint of Mochizuki is that his critics, what he terms “the RCS” (namely, Peter Scholze and Jakob Stix, I don’t know why he doesn’t do them the courtesy of naming them), are doing the former, but he is pointing to the latter as giving rise to some kind of problem. But this metaphor is at best a metaphor: it is not “closely technically related” to IUT, since i) and ii) are dealing with different meanings of the word, applying in different situations. The indented sentence starting “it is by no means…” is completely irrelevant, and I feel confused, since the sense in which \{0\} and \{1\} are identified as being isomorphic in the first half of the sentence is completely different from the sense in which they are identified as being equal points in a quotient space in the second sentence.

If Mochizuki’s writing was less circumlocutory (why do we need fancy notation for basic objects? and reminders of how they are built?), then it would be easier to grasp these apparently illustrative and enlightening examples. But it’s taken me several readings to understand that this seems to be a metaphor, at best, rather than a blatantly false statement. But the metaphor is being conflated with the situation it is meant to be explaining, due to a quirk of terminology. To quote the man himself from section 2.2 of the document “In fact, of course, such “pseudo-mathematical reasoning” is itself fundamentally flawed. [sic]”. I don’t find the pseudo-explanations involving line segments as metaphors for morphisms in a category at all useful, and indeed I find them actively misleading for people who aren’t familiar enough with category theory to take the metaphor of 2. above as being a faithful representation of the situation in 1.

Added: There was some discussion on Reddit about this post. I made some responses around points that I realised I failed I made clearly enough, and now I think my issue with the above example is that it is a total straw man. Part i) could have been written without reference to [0,1] or the specific subspaces \{0\} and \{1\}, and it’s really just talking about terminal objects in a category, and the possibility that one can assume there is a unique terminal object, as long as one is willing to replace the category one is working with an equivalent one (you don’t even need to pass to a skeleton). Reusing the notation and terminology from i) in ii) seems chosen to conflate the two in the mind of the casual reader (it did for me, and I know better!). The indented sentence could be phrased more accurately and transparently as

“the fact all terminal objects in a category are uniquely isomorphic to each other doesn’t imply taking a coequaliser won’t give you a new object”

which is true, but tells us nothing about the veracity of Scholze and Stix’s argument.

13 thoughts on “This argument of Mochizuki doesn’t make sense to me

  1. Sorry, I think you are wrong in saying that, “as all points are identified, then any space collapses to one point also” precisely because the skeletal category of topological spaces also contains its morphisms.

    That is why I is identified with L in the skeleton cathegory, because in the skeletal category, the notion of “interval” cannot exist, as it implies the existence of two different (albeit isomorphic) “points”. He is speaking of the category of simplices, as far as I understand.

    In that skeletal category you have morphisms

    {\alpha}->L

    but not

    L->{\alpha}

    Mochizuki’s example is correct in the sense that identifying objects (essentially all k-dimensional spaces —I know this is not so, but bear with me—) may perfectly limit the reasoning because the “quotient” category may have much less information (objects and maps) than the original one.

    But I am playing Devil’s advocate here.

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    1. Oh, I absolutely know that the argument doesn’t work. Of course the skeletal category contains all the appropriate morphisms.

      I is not identified with L in the skeletal category, and you can absolutely have different points while only having a single terminal object. A skeletal category of finite sets does not mean that combinatorics collapses, after all 😉

      I’m not sure what you mean about the category of simplices, unless you mean the subcategory of Top on the standard topological simplices.

      As far as I know, you definitely can have a morphism from an interval to a point.

      And there’s been nothing said here about quotient categories, but skeleta, which are equivalent full subcategories. One can get a retraction to a skeleton with a suitable amount of Choice, but this retraction is still fully faithful, so shouldn’t be considered as a quotient.

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      1. Yes, I was just trying to make some sense of something very strange (but it is rather impossible). Thanks for your response.

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  2. Identifying isomorphic objects in any category results in an isomorphic category.

    Identifying isomorphic subspaces of a topological space need not result in an isomorphic topological space.

    There’s no contradiction here and so I’m not sure what point Mochizuki’s example is supposed to show.

    But I wouldn’t say it’s two different uses of the concept of “identification” so much as it is just two different categories behaving differently. All the talk of skeleta seems rather irrelevant.

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    1. All the talk of skeleta seems rather irrelevant.

      In this case, yes, since the argument would be the same if one just took the full subcategory with only a single terminal object, rather than a skeleton.

      I still hold that Mochizuki is thinking of “identification” in the sense of choosing an isomorphism to a single object in an isomorphism class, this has come up a lot in previous writing, if not in the particular passage in question. But there’s no doubt that the example doesn’t seem relevant.

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  3. As a friend pointed out to me when I raised this objection, the statement in 2.3.1 (i) about skeletal categories seems to just be an offhand remark, having little to do with the example of 2.3.2 (ii). Notice that Mochizuki doesn’t actually use the word “identify” in 2.3.1 (i).

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    1. This is true, but he has used it elsewhere when complaining about Scholze and Stix doing this, as they did themselves. It’s the reason I chose the title “A crisis of identification” for my 2019 essay on the matter.

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  4. “I don’t know why he doesn’t do them the courtesy of naming them”

    My guess: to focus on ideas rather than persons.

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    1. Sure, and Mochizuki explicitly says this. But one suspects there is more at play. Someone pointed out elsewhere it might be cultural, to be more polite by avoiding naming them while saying they are wrong. Seems plausible to me.

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      1. The practical effect of this “polite” action is that Grothendieck’s SGA gets cited, but Scholze-Stix is missing from the bibliography. Mochizuki knew exactly what he was doing.

        The argument from culture is really disingenuous in this context. He isn’t some Japanese Ramanujan who’s just been discovered by his Hardy. He grew up in America, and studied and worked there before joining RIMS, so the argument from culture is really an attempt at erasing his lived experience as a Japanese abroad.

        Finally, if we’re really serious about focusing on ideas rather than persons, then we should stop insisting on having bibliographies in publications and naming mathematical objects after people:

        https://nautil.us/issue/89/the-dark-side/why-mathematicians-should-stop-naming-things-after-each-other

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      2. I agree that letting M off the hook for every single thing that seems odd for reasons of culture is a non-starter. We are all just reading the increasingly obscure tea-leaves here, and I certainly am not going to make any definite pronouncements on this.

        “Finally, if we’re really serious about focusing on ideas rather than persons, then we should stop insisting on having bibliographies in publications and naming mathematical objects after people:”

        And IUT should not have been named after Teichmüller, if people’s names shouldn’t be brought into it!

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      3. “And IUT should not have been named after Teichmüller, if people’s names shouldn’t be brought into it!”

        Would it have been better if Mochizuki had heeded that and called it “M-theory” instead? 🙂

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